Mathematics Awareness Month Theme Essay

The Shape of the Universe

Early Models of the Shape of
the Universe

For thousands of years, people
believed that the universe revolved around the earth
and astronomers created mathematical models in order
to explain observations in the sky. Eudoxus created
a model containing rotating spheres centered about the
earth. Aristotle adopted
and described this model. While he was able to partially
explain some of the planetary motions by rotating the
spheres at different velocities, other observations,
such as differences in brightness levels, could not be
resolved. In his famous work, the Almagest, Claudius
Ptolemy, a 2nd century philosopher, refined and improved
the model based on the earlier work of Apollonius and Hipparchus.
In the Ptolemaic Universe planets now moved along
epicycles (see the related JavaSketch for
more information [8]), which had circles
attached to the spheres around the earth, and yet this
model still did not completely resolve earlier difficulties.

Aristarchus had
suggested a heliocentric system and in the 16th century, Nicolaus
Copernicus gave substance to Aristarchus' ideas by
carrying out the detailed mathematical calculations:

Finally we shall place
the Sun himself at the center of the Universe. All
this is suggested by the systematic procession of
events and the harmony of the whole Universe, if
only we face the facts, as they say, 'with both eyes
open'.

Copernicus,
De Revolutionibus Orbium Coelestium, 1543

His model still utilized epicycles
in order to explain the circular motion of the planets,
but it placed a motionless sun close to the center of
the universe.

Johannes
Kepler revolutionized astronomy by finally overthrowing
the stranglehold of purely circular motions. His introduction
of elliptical orbits, together with his other two laws
of planetary motion, form the basis of celestial mechanics
to this day. They were also critical in the formulation
and verification of Sir
Isaac Newton's laws of gravity and of motion, which
in turn became the basis for cosmology for the following
two centuries.

In 1917, Albert
Einstein used Bernhard
Riemann's mathematics in order to present a model
for the universe that was consistent with his theory
of relativity. His model was based on a finite spherical
universe that obeyed different laws than those of a
flat Euclidean universe [6]. Since
then mathematicians and astronomers have debated whether
the universe is Euclidean or non-Euclidean, and whether
it is finite or infinite.

Euclidean or Non-Euclidean?

Carl
Friedrich Gauss performed an experiment to measure
the angles of a triangle formed by light rays whose
vertices were between Brocken, Hohenhagen, and Inselberg,
three mountain peaks in Germany. Since Euclidean triangles
have 180 degrees as the measure of the sum of the angles
(see the related JavaSketch for
more information), it might seem that this method could
determine whether the universe is Euclidean. Even if
the universe is non-Euclidean, a triangle of this size
would have been too small to detect the possible curvature
within the accuracy of the measuring instruments, and
light rays are not good measures of shortest distance
paths because they bend with gravity. As far as we
know, while it is true that Gauss did perform measurements
on this triangle, it is a myth that he wanted to use
these measurements to determine whether the universe
is Euclidean [1]. Instead, it was Nikolai
Lobachevsky who proposed to measure the triangle
formed by the star Sirius and the position of the earth
at different times, 6 months apart, in order to determine
the curvature of space [7]. The Sirius
measurements were inconclusive because they were close
to 180 degrees, within the expected margin of error.

In 1900, Karl
Schwarzschild conducted a detailed analysis of
the amount the universe would have to be positively
or negatively curved in order for the known parallax
measurements at the time to be Euclidean within the
margin of error then available. Today, astronomers
still wonder whether our universe is Euclidean or not.
In the foreseeable future, we can only measure triangles
with an area less than that of our solar system, but
we would need to look further than that to determine
the geometry [6].

Finite or Infinite?

There is also disagreement over
whether the universe is finite or infinite. Astronomers
have noticed a cosmic microwave background radiation
coming from all directions of space. There are slight
variations that may enable us to determine the shape
of space, although none of the observations so far have
been accurate enough to make a definite determination.
Recent probes such as NASA's WMAP (Wilkinson Microwave
Anisotropy Probe) and a planned 2007 launch of the Planck
satellite hope to do so.

Astronomers have analyzed the
WMAP data and they have obtained conflicting results.
Jean-Pierre Luminet and his colleagues proposed that
the data seemed to best fit a universe that was a spherical
space formed by identifying opposite faces of a dodecahedron
in a three-dimensional sphere [10].
You can build a dodecahedron, a polyhedron with 12 pentagonal
faces, to see that the faces cannot be glued straight
across without first using a twist. Other mathematicians
and physicists, such as Max Tegmark and his colleagues,
assert that the WMAP data in fact rules out a finite
universe, and that measurements point to a flat Euclidean
space which is infinite [11].

Understanding the Finite Theory

Our universe could be finite
but still have no edges. The first to make that observation
was Riemann, with his proposal of a spherical universe.
As another example, a spaceship traveling off one side
of the screen below will reappear on the opposite side,
creating the illusion of traveling in an infinite space.
To see what finite shape this actually is, you can start
with a square piece of stretchy material and glue together
the left and right sides to obtain a cylinder. Next,
identify the top and bottom sides and the cylinder will
fold up into a torus. In a similar way, identify opposite
faces of a cube or a polyhedron in higher dimensions
to obtain a possible shape for our universe [14].

You can experience what it is
like to live on a 2-D torus universe by playing Torus
Tic-Tac-Toe[12]. In the game,
the top left square is really next to the top right square.
You are allowed to "scroll" the board in order
to help develop your intuition (once a square has been
labeled X or O, you can click on it, hold it down, and
move the board around to see the identifications).

To visualize a finite universe
in three dimensions, you can watch the Futurama[4] episode I,
Roommate (Season 1 DVD):

Fry and Bender are looking
for housing. Leela, Fry, Bender and the manager enter
an apartment that resembles Dutch graphic artist M.C.
Escher's Relativity print [3]. Fry: I'm not sure we wanna pay for a dimension
we're not gonna use.
Bender, the robot, falls down the staircase and continues
to fall "down" the other staircases in many
different directions.

Look for Bender in each of the
frames, and use his position to give gluing instructions
and explain which openings are identified [5].

Regardless of whether our universe
is finite or infinite, the classification of finite universes
is an area of intense interest to mathematicians. There
are ten closed three-dimensional Euclidean universes,
glued together by identifying the opposite faces of a
cube, just like the above Futurama apartment.
The number of spherical possibilities are infinite, but
they have been classified completely, and one of them
is the dodecahedral universe mentioned by Luminet. There
are infinitely many possibilities for a hyperbolic universe,
another non-Euclidean geometry [2],
and their rich structure and classification is still
the subject of research today.

Conclusion

The quest to understand the precise
geometry and shape of our universe began thousands of
years ago, when mathematicians and astronomers used mathematical
models to try and explain their observations. While there
seem to be some irregularities in the WMAP data that
throw the conflicting analyses and conclusions into doubt [13],
there is hope that the data from the proposed 2007 Planck
satellite will ultimately lead us to the answer. Even
if it does not, we will continue to develop new models
and methods so that one day we can determine the shape
of space.

The author gratefully
acknowledges the input of Robert Osserman of MSRI.